Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Suppose that the graph of \(y=x^{2}\) is translated in such a way that its domain is \((-\infty, \infty)\) and its range is \([38, \infty)\). What values of \(h\) and \(k\) can be used if the new function is of the form \(y=(x-h)^{2}+k ?\) (Graph cannot copy)

Step-by-Step Solution

Answer

The values are \(h = 0\) and \(k = 38\).

Step 1: Understanding Translations

When a graph of a function is translated, its basic shape doesn't change, but its position does. For the function \(y = x^2\), a translation can be vertical, represented by \( y = x^2 + k \), or horizontal, represented by \( y = (x-h)^2 \). Here, the translation results in a new range \([38, \infty)\).

Step 2: Analyzing the Vertex Change

The vertex of the function \(y = x^2\) is at \((0, 0)\), and the range is \([0, \infty)\). We need the range to be \([38, \infty)\), indicating the vertex must shift up to \( (h, k) = (0, 38)\). This means the function is vertically translated \(38\) units upward.

Step 3: Identifying Translation Parameters

The horizontal translation \(h\) doesn't affect the range but only shifts the graph horizontally. Since there's no stipulation for changing the domain or horizontal shift in the problem, \(h\) remains \(0\). The achieved range by vertical shift confirms \(k = 38\).

Step 4: Writing the Translated Function

The function considering the translations is \(y = (x-0)^2 + 38 \), which simplifies to \(y = x^2 + 38\). This new function correctly translates the vertex and adjusts the range appropriately.

Key Concepts

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in algebra and they are typically represented by the equation \( y = ax^2 + bx + c \). When we talk about quadratic functions, the most basic form and the one most frequently examined is \( y = x^2 \). This form represents a parabola that opens upward with its vertex at the origin \( (0, 0) \).
Quintessential features of quadratic functions include:

• Vertex: The highest or lowest point of the parabola, depending on whether it opens upward or downward.
• Axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• Opening direction: Determined by the coefficient \( a \). If \( a > 0 \), the parabola opens upward; if \( a < 0 \), it opens downward.
• Transformations: Include shifting, reflecting, stretching, and compressing the graph, usually involving changes to coefficients in the function's equation.

Understanding these core properties helps in comprehending how translations or shifts affect the position and shape of the graph.

Exploring Vertex Shifts in Quadratic Functions

A vertex shift refers to the movement of the vertex of a quadratic function's graph, typically described by changes in parameters \( h \) and \( k \) in the vertex form equation \( y = (x-h)^2 + k \).
Here, the variable \( h \) dictates the horizontal shift, while \( k \) dictates the vertical shift. When the vertex of the standard parabola \( y = x^2 \) moves to a new position, both the horizontal and vertical location of the parabola also change.

• Vertical Shifts: If \( k > 0 \), the parabola shifts up; if \( k < 0 \), it shifts down, changing the range accordingly.
• Horizontal Shifts: Adjustments in \( h \) translate the graph left or right. Specifically, \( (x-h) \) shifts right if \( h > 0 \) and left if \( h < 0 \).

In the original exercise, determining the range \([38, \infty)\) required identifying that there was no horizontal shift (\( h = 0 \)) and a significant upward vertical shift (\( k = 38 \)). These shifts, essential in modifying graph positions, are transformative properties inherent to quadratic functions.

Understanding the Range of a Quadratic Function

The range of a function is the set of all possible output values. For a basic quadratic function \( y = x^2 \), the range is \([0, \infty)\), because the smallest value \( y \) can take is 0, increasing as \( x \) moves away from the origin in either direction.
When we translate a parabola vertically, as seen with our function \( y = x^2 + 38 \), the whole graph shifts up by 38 units adjusting the minimum point of the parabola.
Thus:

• The new range becomes \([38, \infty)\).
• This upward translation doesn't affect the domain, which remains as \(( -\infty, \infty )\).

The range adjustment through vertical translation allows us to control where the functions output starts, which is especially useful in applications requiring function values to begin at a specific low or high point. Understanding how translations affect the vertex and range is critical for analyzing and manipulating quadratic functions.